\(\int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx\) [55]

Optimal result

Integrand size = 26, antiderivative size = 157 \[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {8 b \sqrt {\pi } x}{105 c^5}+\frac {4 b \sqrt {\pi } x^3}{315 c^3}-\frac {b \sqrt {\pi } x^5}{175 c}-\frac {1}{49} b c \sqrt {\pi } x^7+\frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c^6 \pi }-\frac {2 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^6 \pi ^2}+\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^6 \pi ^3} \] Output:

-8/105*b*Pi^(1/2)*x/c^5+4/315*b*Pi^(1/2)*x^3/c^3-1/175*b*Pi^(1/2)*x^5/c-1/ 
49*b*c*Pi^(1/2)*x^7+1/3*(Pi*c^2*x^2+Pi)^(3/2)*(a+b*arcsinh(c*x))/c^6/Pi-2/ 
5*(Pi*c^2*x^2+Pi)^(5/2)*(a+b*arcsinh(c*x))/c^6/Pi^2+1/7*(Pi*c^2*x^2+Pi)^(7 
/2)*(a+b*arcsinh(c*x))/c^6/Pi^3
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.84 \[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi } \left (105 a \sqrt {1+c^2 x^2} \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right )-b c x \left (840-140 c^2 x^2+63 c^4 x^4+225 c^6 x^6\right )+105 b \sqrt {1+c^2 x^2} \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right ) \text {arcsinh}(c x)\right )}{11025 c^6} \] Input:

Integrate[x^5*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]),x]
 

Output:

(Sqrt[Pi]*(105*a*Sqrt[1 + c^2*x^2]*(8 - 4*c^2*x^2 + 3*c^4*x^4 + 15*c^6*x^6 
) - b*c*x*(840 - 140*c^2*x^2 + 63*c^4*x^4 + 225*c^6*x^6) + 105*b*Sqrt[1 + 
c^2*x^2]*(8 - 4*c^2*x^2 + 3*c^4*x^4 + 15*c^6*x^6)*ArcSinh[c*x]))/(11025*c^ 
6)
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6219, 27, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^5 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x)) \, dx\)

\(\Big \downarrow \) 6219

\(\displaystyle -\sqrt {\pi } b c \int \frac {15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8}{105 c^6}dx+\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi c^6}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\sqrt {\pi } b \int \left (15 c^6 x^6+3 c^4 x^4-4 c^2 x^2+8\right )dx}{105 c^5}+\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi c^6}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi ^3 c^6}-\frac {2 \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))}{5 \pi ^2 c^6}+\frac {\left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))}{3 \pi c^6}-\frac {\sqrt {\pi } b \left (\frac {15 c^6 x^7}{7}+\frac {3 c^4 x^5}{5}-\frac {4 c^2 x^3}{3}+8 x\right )}{105 c^5}\)

Input:

Int[x^5*Sqrt[Pi + c^2*Pi*x^2]*(a + b*ArcSinh[c*x]),x]
 

Output:

-1/105*(b*Sqrt[Pi]*(8*x - (4*c^2*x^3)/3 + (3*c^4*x^5)/5 + (15*c^6*x^7)/7)) 
/c^5 + ((Pi + c^2*Pi*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*c^6*Pi) - (2*(Pi 
+ c^2*Pi*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^6*Pi^2) + ((Pi + c^2*Pi*x^2 
)^(7/2)*(a + b*ArcSinh[c*x]))/(7*c^6*Pi^3)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6219

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_ 
), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^2)^p, x]}, Simp[(a + b*ArcSi 
nh[c*x])   u, x] - Simp[b*c*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[S 
implifyIntegrand[u/Sqrt[d + e*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x 
] && EqQ[e, c^2*d] && IntegerQ[p - 1/2] && NeQ[p, -2^(-1)] && (IGtQ[(m + 1) 
/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
 

Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.20

method	result	size
orering	\(\frac {\left (2925 c^{8} x^{8}+3393 c^{6} x^{6}-630 c^{4} x^{4}+4760 c^{2} x^{2}+5040\right ) \sqrt {\pi \,c^{2} x^{2}+\pi }\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}{11025 c^{6} \left (c^{2} x^{2}+1\right )}-\frac {\left (225 c^{6} x^{6}+63 c^{4} x^{4}-140 c^{2} x^{2}+840\right ) \left (5 x^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }\, \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )+\frac {x^{6} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) \pi \,c^{2}}{\sqrt {\pi \,c^{2} x^{2}+\pi }}+\frac {x^{5} \sqrt {\pi \,c^{2} x^{2}+\pi }\, b c}{\sqrt {c^{2} x^{2}+1}}\right )}{11025 x^{4} c^{6}}\)	\(189\)
default	\(a \left (\frac {x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{7 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{5 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{15 \pi \,c^{4}}\right )}{7 c^{2}}\right )+\frac {b \sqrt {\pi }\, \left (1575 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}+1890 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-225 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}-105 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-63 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+420 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+140 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+840 \,\operatorname {arcsinh}\left (x c \right )-840 \sqrt {c^{2} x^{2}+1}\, x c \right )}{11025 c^{6} \sqrt {c^{2} x^{2}+1}}\)	\(224\)
parts	\(a \left (\frac {x^{4} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{7 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{5 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{15 \pi \,c^{4}}\right )}{7 c^{2}}\right )+\frac {b \sqrt {\pi }\, \left (1575 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}+1890 \,\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}-225 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}-105 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-63 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+420 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+140 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+840 \,\operatorname {arcsinh}\left (x c \right )-840 \sqrt {c^{2} x^{2}+1}\, x c \right )}{11025 c^{6} \sqrt {c^{2} x^{2}+1}}\)	\(224\)

Input:

int(x^5*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(x*c)),x,method=_RETURNVERBOSE)
 

Output:

1/11025*(2925*c^8*x^8+3393*c^6*x^6-630*c^4*x^4+4760*c^2*x^2+5040)/c^6/(c^2 
*x^2+1)*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(x*c))-1/11025/x^4*(225*c^6*x^6+ 
63*c^4*x^4-140*c^2*x^2+840)/c^6*(5*x^4*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh( 
x*c))+x^6/(Pi*c^2*x^2+Pi)^(1/2)*(a+b*arcsinh(x*c))*Pi*c^2+x^5*(Pi*c^2*x^2+ 
Pi)^(1/2)*b*c/(c^2*x^2+1)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.18 \[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {105 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (15 \, b c^{8} x^{8} + 18 \, b c^{6} x^{6} - b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (1575 \, a c^{8} x^{8} + 1890 \, a c^{6} x^{6} - 105 \, a c^{4} x^{4} + 420 \, a c^{2} x^{2} - {\left (225 \, b c^{7} x^{7} + 63 \, b c^{5} x^{5} - 140 \, b c^{3} x^{3} + 840 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 840 \, a\right )}}{11025 \, {\left (c^{8} x^{2} + c^{6}\right )}} \] Input:

integrate(x^5*(pi*c^2*x^2+pi)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="frica 
s")
 

Output:

1/11025*(105*sqrt(pi + pi*c^2*x^2)*(15*b*c^8*x^8 + 18*b*c^6*x^6 - b*c^4*x^ 
4 + 4*b*c^2*x^2 + 8*b)*log(c*x + sqrt(c^2*x^2 + 1)) + sqrt(pi + pi*c^2*x^2 
)*(1575*a*c^8*x^8 + 1890*a*c^6*x^6 - 105*a*c^4*x^4 + 420*a*c^2*x^2 - (225* 
b*c^7*x^7 + 63*b*c^5*x^5 - 140*b*c^3*x^3 + 840*b*c*x)*sqrt(c^2*x^2 + 1) + 
840*a))/(c^8*x^2 + c^6)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (150) = 300\).

Time = 3.07 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.92 \[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\sqrt {\pi } a x^{6} \sqrt {c^{2} x^{2} + 1}}{7} + \frac {\sqrt {\pi } a x^{4} \sqrt {c^{2} x^{2} + 1}}{35 c^{2}} - \frac {4 \sqrt {\pi } a x^{2} \sqrt {c^{2} x^{2} + 1}}{105 c^{4}} + \frac {8 \sqrt {\pi } a \sqrt {c^{2} x^{2} + 1}}{105 c^{6}} - \frac {\sqrt {\pi } b c x^{7}}{49} + \frac {\sqrt {\pi } b x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {\sqrt {\pi } b x^{5}}{175 c} + \frac {\sqrt {\pi } b x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{35 c^{2}} + \frac {4 \sqrt {\pi } b x^{3}}{315 c^{3}} - \frac {4 \sqrt {\pi } b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{105 c^{4}} - \frac {8 \sqrt {\pi } b x}{105 c^{5}} + \frac {8 \sqrt {\pi } b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{105 c^{6}} & \text {for}\: c \neq 0 \\\frac {\sqrt {\pi } a x^{6}}{6} & \text {otherwise} \end {cases} \] Input:

integrate(x**5*(pi*c**2*x**2+pi)**(1/2)*(a+b*asinh(c*x)),x)
 

Output:

Piecewise((sqrt(pi)*a*x**6*sqrt(c**2*x**2 + 1)/7 + sqrt(pi)*a*x**4*sqrt(c* 
*2*x**2 + 1)/(35*c**2) - 4*sqrt(pi)*a*x**2*sqrt(c**2*x**2 + 1)/(105*c**4) 
+ 8*sqrt(pi)*a*sqrt(c**2*x**2 + 1)/(105*c**6) - sqrt(pi)*b*c*x**7/49 + sqr 
t(pi)*b*x**6*sqrt(c**2*x**2 + 1)*asinh(c*x)/7 - sqrt(pi)*b*x**5/(175*c) + 
sqrt(pi)*b*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(35*c**2) + 4*sqrt(pi)*b*x* 
*3/(315*c**3) - 4*sqrt(pi)*b*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(105*c**4 
) - 8*sqrt(pi)*b*x/(105*c**5) + 8*sqrt(pi)*b*sqrt(c**2*x**2 + 1)*asinh(c*x 
)/(105*c**6), Ne(c, 0)), (sqrt(pi)*a*x**6/6, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.22 \[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{105} \, {\left (\frac {15 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{4}}{\pi c^{2}} - \frac {12 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{2}}{\pi c^{4}} + \frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}{\pi c^{6}}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{105} \, {\left (\frac {15 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{4}}{\pi c^{2}} - \frac {12 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} x^{2}}{\pi c^{4}} + \frac {8 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}}}{\pi c^{6}}\right )} a - \frac {{\left (225 \, \sqrt {\pi } c^{6} x^{7} + 63 \, \sqrt {\pi } c^{4} x^{5} - 140 \, \sqrt {\pi } c^{2} x^{3} + 840 \, \sqrt {\pi } x\right )} b}{11025 \, c^{5}} \] Input:

integrate(x^5*(pi*c^2*x^2+pi)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="maxim 
a")
 

Output:

1/105*(15*(pi + pi*c^2*x^2)^(3/2)*x^4/(pi*c^2) - 12*(pi + pi*c^2*x^2)^(3/2 
)*x^2/(pi*c^4) + 8*(pi + pi*c^2*x^2)^(3/2)/(pi*c^6))*b*arcsinh(c*x) + 1/10 
5*(15*(pi + pi*c^2*x^2)^(3/2)*x^4/(pi*c^2) - 12*(pi + pi*c^2*x^2)^(3/2)*x^ 
2/(pi*c^4) + 8*(pi + pi*c^2*x^2)^(3/2)/(pi*c^6))*a - 1/11025*(225*sqrt(pi) 
*c^6*x^7 + 63*sqrt(pi)*c^4*x^5 - 140*sqrt(pi)*c^2*x^3 + 840*sqrt(pi)*x)*b/ 
c^5
 

Giac [F(-2)]

Exception generated. \[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^5*(pi*c^2*x^2+pi)^(1/2)*(a+b*arcsinh(c*x)),x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\int x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {\Pi \,c^2\,x^2+\Pi } \,d x \] Input:

int(x^5*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2),x)
 

Output:

int(x^5*(a + b*asinh(c*x))*(Pi + Pi*c^2*x^2)^(1/2), x)
 

Reduce [F]

\[ \int x^5 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {\pi }\, \left (15 \sqrt {c^{2} x^{2}+1}\, a \,c^{6} x^{6}+3 \sqrt {c^{2} x^{2}+1}\, a \,c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, a \,c^{2} x^{2}+8 \sqrt {c^{2} x^{2}+1}\, a +105 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{5}d x \right ) b \,c^{6}\right )}{105 c^{6}} \] Input:

int(x^5*(Pi*c^2*x^2+Pi)^(1/2)*(a+b*asinh(c*x)),x)
 

Output:

(sqrt(pi)*(15*sqrt(c**2*x**2 + 1)*a*c**6*x**6 + 3*sqrt(c**2*x**2 + 1)*a*c* 
*4*x**4 - 4*sqrt(c**2*x**2 + 1)*a*c**2*x**2 + 8*sqrt(c**2*x**2 + 1)*a + 10 
5*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**5,x)*b*c**6))/(105*c**6)